for all \(1\le p\le \infty \)  and \(d\ge 3\) . The fact that
these operators exist and are isometries \(L^2\rightarrow \mathrm {Ran}(P_c(H))\)  is a classical fact, see Kato [1]}.
They intertwine the free evolution with that of \(H\)  in the sense that (with \(H_0=-\Delta \) )
\(f(H)P_c(H) = W f(H_0) W^*\) 
